Theorem eulerthlem1 12792

Description: Lemma for eulerth 12798. (Contributed by Mario Carneiro, 8-May-2015.)

Hypotheses

Ref	Expression
eulerthlem1.1	|- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
eulerthlem1.2	|- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
eulerthlem1.3	|- T = ( 1 ... ( phi ` N ) )
eulerthlem1.4	|- ( ph -> F : T -1-1-onto-> S )
eulerthlem1.5	|- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )

Assertion

Ref	Expression
eulerthlem1	|- ( ph -> G : T --> S )

Distinct variable groups:   x, y, A   x, F, y   x, G, y   x, N, y   x, S   ph, x, y   x, T, y

Allowed substitution hint:   S( y)

Proof of Theorem eulerthlem1

Step	Hyp	Ref	Expression
1		eulerthlem1.1	. . . . . . 7 |- ( ph -> ( N e. NN /\ A e. ZZ /\ ( A gcd N ) = 1 ) )
2	1	simp2d 1034	. . . . . 6 |- ( ph -> A e. ZZ )
3	2	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> A e. ZZ )
4		eulerthlem1.4	. . . . . . . . . 10 |- ( ph -> F : T -1-1-onto-> S )
5		f1of 5580	. . . . . . . . . 10 |- ( F : T -1-1-onto-> S -> F : T --> S )
6	4, 5	syl 14	. . . . . . . . 9 |- ( ph -> F : T --> S )
7	6	ffvelcdmda 5778	. . . . . . . 8 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. S )
8		oveq1 6020	. . . . . . . . . 10 |- ( y = ( F ` x ) -> ( y gcd N ) = ( ( F ` x ) gcd N ) )
9	8	eqeq1d 2238	. . . . . . . . 9 |- ( y = ( F ` x ) -> ( ( y gcd N ) = 1 <-> ( ( F ` x ) gcd N ) = 1 ) )
10		eulerthlem1.2	. . . . . . . . 9 |- S = { y e. ( 0..^ N ) | ( y gcd N ) = 1 }
11	9, 10	elrab2 2963	. . . . . . . 8 |- ( ( F ` x ) e. S <-> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
12	7, 11	sylib 122	. . . . . . 7 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) e. ( 0..^ N ) /\ ( ( F ` x ) gcd N ) = 1 ) )
13	12	simpld 112	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ( 0..^ N ) )
14		elfzoelz 10375	. . . . . 6 |- ( ( F ` x ) e. ( 0..^ N ) -> ( F ` x ) e. ZZ )
15	13, 14	syl 14	. . . . 5 |- ( ( ph /\ x e. T ) -> ( F ` x ) e. ZZ )
16	3, 15	zmulcld 9601	. . . 4 |- ( ( ph /\ x e. T ) -> ( A x. ( F ` x ) ) e. ZZ )
17	1	simp1d 1033	. . . . 5 |- ( ph -> N e. NN )
18	17	adantr 276	. . . 4 |- ( ( ph /\ x e. T ) -> N e. NN )
19		zmodfzo 10602	. . . 4 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
20	16, 18, 19	syl2anc 411	. . 3 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) )
21		modgcd 12555	. . . . 5 |- ( ( ( A x. ( F ` x ) ) e. ZZ /\ N e. NN ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
22	16, 18, 21	syl2anc 411	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = ( ( A x. ( F ` x ) ) gcd N ) )
23	17	nnzd 9594	. . . . . 6 |- ( ph -> N e. ZZ )
24	23	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> N e. ZZ )
25	16, 24	gcdcomd 12538	. . . 4 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) gcd N ) = ( N gcd ( A x. ( F ` x ) ) ) )
26	23, 2	gcdcomd 12538	. . . . . . 7 |- ( ph -> ( N gcd A ) = ( A gcd N ) )
27	1	simp3d 1035	. . . . . . 7 |- ( ph -> ( A gcd N ) = 1 )
28	26, 27	eqtrd 2262	. . . . . 6 |- ( ph -> ( N gcd A ) = 1 )
29	28	adantr 276	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd A ) = 1 )
30	24, 15	gcdcomd 12538	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = ( ( F ` x ) gcd N ) )
31	12	simprd 114	. . . . . 6 |- ( ( ph /\ x e. T ) -> ( ( F ` x ) gcd N ) = 1 )
32	30, 31	eqtrd 2262	. . . . 5 |- ( ( ph /\ x e. T ) -> ( N gcd ( F ` x ) ) = 1 )
33		rpmul 12663	. . . . . 6 |- ( ( N e. ZZ /\ A e. ZZ /\ ( F ` x ) e. ZZ ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
34	24, 3, 15, 33	syl3anc 1271	. . . . 5 |- ( ( ph /\ x e. T ) -> ( ( ( N gcd A ) = 1 /\ ( N gcd ( F ` x ) ) = 1 ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 ) )
35	29, 32, 34	mp2and 433	. . . 4 |- ( ( ph /\ x e. T ) -> ( N gcd ( A x. ( F ` x ) ) ) = 1 )
36	22, 25, 35	3eqtrd 2266	. . 3 |- ( ( ph /\ x e. T ) -> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 )
37		oveq1 6020	. . . . 5 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( y gcd N ) = ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) )
38	37	eqeq1d 2238	. . . 4 |- ( y = ( ( A x. ( F ` x ) ) mod N ) -> ( ( y gcd N ) = 1 <-> ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
39	38, 10	elrab2 2963	. . 3 |- ( ( ( A x. ( F ` x ) ) mod N ) e. S <-> ( ( ( A x. ( F ` x ) ) mod N ) e. ( 0..^ N ) /\ ( ( ( A x. ( F ` x ) ) mod N ) gcd N ) = 1 ) )
40	20, 36, 39	sylanbrc 417	. 2 |- ( ( ph /\ x e. T ) -> ( ( A x. ( F ` x ) ) mod N ) e. S )
41		eulerthlem1.5	. 2 |- G = ( x e. T |-> ( ( A x. ( F ` x ) ) mod N ) )
42	40, 41	fmptd 5797	1 |- ( ph -> G : T --> S )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200   {crab 2512   |-> cmpt 4148   -->wf 5320   -1-1-onto->wf1o 5323   ` cfv 5324  (class class class)co 6013   0cc0 8025   1c1 8026   x. cmul 8030   NNcn 9136   ZZcz 9472   ...cfz 10236  ..^cfzo 10370   mod cmo 10577   gcd cgcd 12517   phicphi 12774

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-sup 7177  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-fz 10237  df-fzo 10371  df-fl 10523  df-mod 10578  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553  df-dvds 12342  df-gcd 12518

This theorem is referenced by:  eulerthlemh  12796  eulerthlemth  12797